Bilinear Transform Simulator — Analog to Digital IIR
Convert an analog first-order lowpass filter H(s)=ωc/(s+ωc) into a digital IIR filter with the bilinear transform. Change the cutoff and sampling frequency to see the IIR coefficients b0/b1/a1, the frequency response, the frequency warping and the effect of pre-warping update in real time.
Parameters
Analog design cutoff fc
Hz
The target −3 dB frequency of the lowpass
Sampling frequency fs
Hz
Sampling period is T = 1/fs; fs/2 is the Nyquist frequency
Evaluation frequency fEval
Hz
The frequency at which to read the digital gain |H|
Results
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Pre-warped ωc (rad/s)
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Coefficient b0
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Coefficient b1
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Coefficient a1
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Gain |H| at fEval (dB)
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Frequency warping ratio
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s-plane → z-plane mapping
Left: the analog s-plane (jω-axis and the left half-plane = the stable region). Right: the digital z-plane (the unit circle and its interior = the stable region). The bilinear transform maps the s left half-plane into the interior of the unit circle. The marked point in the z-plane is the IIR filter's pole (inside the circle = stable).
Digital IIR coefficients obtained by transforming the analog first-order lowpass H(s)=ωc/(s+ωc). The difference equation is y[n] = b0·x[n] + b1·x[n−1] − a1·y[n−1]. The bilinear transform is stable-preserving: the s left half-plane maps inside the z unit circle.
What is the Bilinear Transform?
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The "bilinear transform" turns an analog filter into a digital one, right? But why do we even need a conversion — why not just design directly in the digital domain?
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Good question. The reason is simple: a century of analog filter design theory has piled up. Butterworth, Chebyshev, elliptic — each is a complete theory worked out in the "s-plane". We would rather reuse all of that. So the idea is "design in analog, then translate to digital at the very end". The definitive translator for that step is the bilinear transform. One single substitution, s = (2/T)(1−z⁻¹)/(1+z⁻¹), turns an analog H(s) into a digital H(z).
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So it is a translator. But translation can lose nuance, can't it? Does the bilinear transform distort anything?
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Sharp — that is exactly the catch. The bilinear transform takes the analog frequency axis (0 to infinity) and "crams" it into the finite range 0 to fs/2 in the digital domain. Squeezing the infinite into the finite has to distort something. That distortion is "frequency warping". At low frequencies there is almost no shift, but the closer you get to the Nyquist frequency fs/2, the more the analog frequency is compressed. Look at the "Frequency warping" chart above: the low end is a straight line, but the high end bends over hard.
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So the cutoff frequency I designed gets shifted a bit too? Like, I design for 1000 Hz but the digital filter ends up at 980 Hz?
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Exactly — left alone, it shifts. But there is a proper fix: "pre-warping". If you know warping will shrink it, just stretch it in advance by the amount it will shrink. ωcWarped = (2/T)·tan(ωc·T/2) stretches the analog design cutoff ahead of time, so after the transform it lands right on the target fc. Set the sliders to fc=1000, fs=44100 and you will see the pre-warped ωc come out a tiny bit larger than the original 2π×1000. That "tiny bit" is the correction.
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If pre-warping fixes it, is the bilinear transform a flawless, best-in-class method? I have also heard of impulse invariance.
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Less "best-in-class" and more "the practical standard". Impulse invariance samples the analog impulse response directly. It is intuitive, but the s-plane jω-axis folds over itself many times in the z-plane, producing "aliasing". The bilinear transform wraps the whole jω-axis one-to-one onto the unit circle, so aliasing is zero in principle. The only price is frequency warping, and pre-warping erases that too. That is why — for audio EQs or sensor noise removal alike — when you build an IIR filter from an analog prototype, you almost always use the bilinear transform.
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One last thing. Can the digital filter become "unstable" after the conversion and blow up?
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That is another beauty of the bilinear transform — it is mathematically guaranteed to "preserve stability". In the analog world, a filter is stable if its poles lie in the left half of the s-plane. In the digital world, it is stable if its poles lie inside the unit circle of the z-plane. The bilinear transform maps the s left half-plane neatly into the interior of the unit circle. So if your original analog filter is stable, the transformed digital IIR is always stable too. Watch the "s-plane → z-plane" diagram above to see a left-half-plane point land inside the unit circle.
Frequently Asked Questions
The bilinear transform (also called Tustin's method) is a way to convert an analog filter transfer function H(s) into a digital filter H(z). The substitution s = (2/T)(1−z⁻¹)/(1+z⁻¹) maps the entire left half of the s-plane (the stable region) exactly into the interior of the unit circle in the z-plane (the stable region), where T = 1/fs is the sampling period. Because the whole jω-axis of the s-plane wraps onto the unit circle, no aliasing arises from sampling — that is its biggest advantage.
The bilinear transform squeezes the entire analog jω-axis (0 to ∞) onto the digital unit circle (0 to fs/2), so the frequency axis is compressed nonlinearly — this is called frequency warping. As a result, using the analog cutoff directly shifts the digital cutoff. Pre-warping corrects this: ωcWarped = (2/T)·tan(ωc·T/2) stretches the analog design cutoff in advance. When you design with the pre-warped frequency, the digital −3 dB point lands exactly on the target fc.
Substitute the bilinear s into the analog first-order lowpass H(s) = ωc/(s+ωc). With K = ωcWarped·T/2 the result is b0 = b1 = K/(1+K) and a1 = (K−1)/(1+K). The difference equation is y[n] = b0·x[n] + b1·x[n−1] − a1·y[n−1]. For the tool's defaults (fc=1000Hz, fs=44100Hz) this gives b0=b1≈0.0666 and a1≈−0.866, and the gain at the cutoff frequency is exactly −3 dB.
Impulse invariance samples the analog filter's impulse response, but because the s-plane jω-axis folds over in the z-plane it suffers from aliasing. The bilinear transform maps the whole jω-axis one-to-one onto the unit circle, so there is no aliasing and it works for lowpass, highpass and bandpass alike. The price is frequency warping, but that can be corrected at the cutoff with pre-warping. When designing an IIR filter from an analog prototype (Butterworth, etc.), the bilinear transform is almost always the standard choice.
Real-World Applications
Audio equalizers: The EQs in mixers and plug-ins, tone controls and crossover networks are nearly all implemented as IIR filters. Designers first design an analog Butterworth or bell-shaped shelving filter in the s-plane, then drop it to digital coefficients with the bilinear transform. Because audio sampling rates are fixed at 44.1 kHz or 48 kHz, pre-warping the cutoff accurately is directly tied to the precision of the sound.
Sensor signal noise removal: Measurement data from accelerometers, strain gauges and temperature sensors carries high-frequency noise. Running a first- or second-order IIR lowpass on a microcontroller or DSP to smooth it is a classic technique, and this tool's difference equation y[n] = b0·x[n] + b1·x[n−1] − a1·y[n−1] becomes the embedded code directly. Because it achieves comparable attenuation with fewer coefficients and fewer operations than an FIR filter, it is invaluable on resource-limited embedded devices.
Discretizing control systems: The bilinear transform (Tustin's method) is also used when designing a PID controller or lead-lag compensator in analog and implementing it in a digital controller. A continuous-time transfer function can be discretized for the sampling period T, and because stability is preserved, the stability margin of the analog design does not collapse in the digital implementation. It is widely used in servo-motor and inverter firmware.
Communications and biosignal processing: IIR filters are also used in fields that demand real-time operation, such as the channel-selection filter in a radio receiver, baseline-wander removal in an ECG, and band splitting of EEG. Discretizing an analog prototype filter (such as an elliptic filter) with the bilinear transform achieves a sharp roll-off with a low filter order.
Common Misconceptions and Pitfalls
The biggest pitfall is forgetting to pre-warp, so the cutoff shifts. If you plug the analog cutoff ωc directly into K = ωc·T/2 and compute the coefficients, the −3 dB point of the resulting digital filter ends up lower than the target fc. At low frequencies and high sampling rates the shift looks small and is easily overlooked, but the closer the cutoff is to the Nyquist frequency, the less the shift can be ignored. This tool's "frequency warping ratio" is the ratio of the pre-warped ωc to the original 2π·fc. The further this value is from 1, the stronger the warning that without pre-warping the design will not come out as intended.
Next, assuming the bilinear transform preserves both magnitude and phase exactly. What the bilinear transform preserves is the "shape of the magnitude response" and "stability" — the frequency axis itself is warped nonlinearly. In other words, you can match the gain at a given frequency between analog and digital, but the frequency-axis scale is stretched and compressed. The phase response (group delay) in particular will not be the same as the analog prototype. For applications that need linear phase, you should consider an FIR filter rather than an IIR. Understand the bilinear transform as a method that prioritizes "magnitude characteristics and aliasing avoidance".
Finally, computing the coefficients is not the end. This tool finds b0, b1 and a1 with ideal real-number arithmetic, but an actual embedded implementation quantizes the coefficients to a finite number of bits. On fixed-point DSPs in particular, coefficient rounding error can push a pole slightly outside the unit circle, and a filter that should be stable can oscillate. The reason high-order filters are not implemented as a single transfer function but decomposed into a cascade of second-order sections (biquads) is to reduce this quantization sensitivity. The coefficients from the bilinear transform are "design values"; the implementation still needs a separate verification of quantization and numerical precision.
How to Use
Enter analog corner frequency fcNum (Hz) using the slider or text field, ranging from fcRange minimum to maximum.
Set sampling frequency fsNum (Hz) via fsRange to define the digital system's Nyquist limit; typical values: 44.1 kHz (audio), 10 kHz (sensor data).
Specify evaluation frequency feNum (Hz) within feRange to compute gain |H| in dB at that test point.
Click Simulate to compute pre-warped ωc, IIR coefficients (b0, b1, a1), and frequency response magnitude.
Worked Example
Design a 1 kHz analog lowpass filter digitized at fs=48 kHz. Analog transfer function: H(s)=2π·1000/(s+2π·1000). Apply bilinear transform with pre-warping: ωc_warped = 2·fs·tan(ωc·T/2) ≈ 6547 rad/s. Resulting digital coefficients: b0≈0.119, b1≈0.119, a1≈0.762. At evaluation frequency fe=500 Hz: |H|≈−3.01 dB (half-power point). Frequency warping ratio: 1.04× (minimal distortion below 4.8 kHz Nyquist).
Practical Notes
For real-time DSP (embedded systems, microcontrollers): keep fs at least 10× the analog corner frequency to minimize frequency warping and maintain phase linearity.
Audio applications (DAWs, synthesizers) commonly use fs=44.1 or 48 kHz; pre-warping correction prevents the notorious "hard knee" artifact near Nyquist when fc approaches fs/2.
In sensor filtering (accelerometers, strain gauges), oversample at 100× the physical signal bandwidth, then apply this IIR stage to avoid aliasing before analog-to-digital conversion quantization noise.
Monitor warping ratio; values >1.5 indicate significant frequency compression and require re-tuning fc or increasing fs to stay within linear phase response region.